Models of the lunar surface including temperature-dependent thermal properties

~c~aus 5, 606-634 (1966)

Models of the Lunar Surface Including Temperature-Dependent Thermal Properties J E F F R E Y L. L I N S K Y 1 Harvard College Observatory, Cambridge, Massoehusetls Communicated by Carl Sagan Received April 28, 1966 The thermal conditions in the lunar surface are considered on a gross scale in terms of models with temperature-dependent thermal properties, including radiative energy transport. Agreement is obtained with infrared measurements of cold terminator temperatures and radio lunation data at millimeter wavelengths for a range of postulated parameters of the surface material. The observed increase of mean radio brightness temperature with wavelength is interpreted as due to radiative energy transport and the resultant nonlinearity of the heat-conduction equation, rather than to a large radioactive heat flux. The postulated existence of radiative energy transport is consistent with a porous or frothy medium, in agreement with photometric and laboratory simulation experiments, as well as with recent radar depolarization measurements. A distance scale of 0.1-0.3 mm for the effective mean separation of radiating surfaces is suggested by this interpretation ~f the data. only if the conductivity increases with At the present time contradict,ions exist temperature. Recent measurements of postubetween deduced thermal properties of the lated lunar materials indicate such a temlunar surface as interpreted from infrared perature dependence. Therefore, an observed increase of the mean radio brightness temand from radio observations. In addition, high-precision radio observations have been perature with wavelength m a y be indicative interpreted in terms of an unexpectedly large of the thermal properties of the material and not of the net thermal flux resulting from thermal flux in the lunar surface material. In this paper I suggest a means whereby radioactive heating. Secondly, it is argued that radio measurethese contradictions m a y be resolved by ments cannot be interpreted uniquely in ~onsidering temperature dependence of the that two parameters are involved in an conductivity, including a radiative conambiguous manner, i.e., the ratio of the tribution, and of the specific heat. I have thermal to the electromagnetic wavelengths written a computer program to solve the and a product of thermal properties. The heat-conduction equation for arbitrary temradio data may, therefore, be reinterpreted perature- and depth-dependent thermal properties during a Iunation and an eclipse, in terms of models suggested by recent and to compute radio brightness tempera- infrared measurements of the minimum surlures for such models of the lunar surface. face temperature reached during the lunar It is first shown that the mean tempera- night.. Each of eight simple models, six of which ~.ure beneath the surface increases signifihave temperature-dependent thermal propcantly with depth for no net thermal flux erties, agrees well with infrared eclipse and Present address: Smithsonian Astrophysical hmation data as well as with radio observations in the millimeter range. All but one are C)bservatolT, Cambridge, Massachusetts. 606 I. INTRODUCTION

LUNAR SURFACE MODELS

homogeneous with depth to at least several meters and are stratified although a more complex structure is probably more realistic. This group of models suggests that the surface layer may be a highly porous, possibly frothy, one in which radiation plays a significant role in heat transfer in the material. Such a tentative model is in agreement with recent radar depolarization data and laboratory simulation experiments. It should be emphasized that the actual lunar surface may be quite complicated, consisting of many materials arranged in depth and across the surface. This picture is suggested by recent high resolution infrared, radio, and radar data. This paper is an attempt to describe the gross thermal properties of a region of the lunar surface, the subterrestrial region, so as to be consistent with recent infrared, radio, and radar data, by means of simple nmdels incorporating temperature-dependent thermal properties. The approach should give information about this region if the actual inhomogeneities act only as fine structure upon the emitted thermal radiation rather than as important contributions to this radiation. If the contrary proves to be true, which is the more likely case, then the present methodology should be valid provided one weighs the contributions to the infrared and radio signals by the relative proportions of the various materials in the field of view. As the data become more numerous, such an approach becomes more fruitful and should be pursued. I I . PREVIOUS INTERPRETATIONS OF INFRARED AND RADIO

OBSERVATIONS The first attempts at characterizing the thermal properties of the lunar surface consisted of solutions to the heat-conduction equation in a homogeneous, plane-parallel medium of temperature-independent properties under boundary conditions appropriate for a lunation and an eclipse. Wesselink (1948) obtained reasonable agreement between his theoretical eclipse cooling curves and Pettit's (1940) eclipse data for a model in which the thermal parameter ~/ -- (Kpc)-~ ~ 1000 cm" °K cal-~ see-~,

{i07

where K is the thermal conductivity; p, the density; and c, the specific heat. Jaeger and Harper (1950) and some subsequent investigators have obtained better agreement with two-layer models. Prior to 1962 the only temperature measurement of the dark side of the Moon available was the value of 120° 4- 15°K obtained by Pettit and Nicholson (1930), later confirmed by Sinton (1962). Jaeger (1953) found this midnight temperature consistent with a homogeneous model with ~, ~ 500, or by two-layer models consisting of less than a centimeter of insulating materials over a more conducting medium. With the vast improvement of infrared instrumentation-and especially the recent measurements by Low (1965)--the mean temperature of the cold lunar limb has been more reliably established at 90°K. Thus both eclipse and lunation infrared measurements may be explained in terms of a typical lunar surface element homogeneous at least to a depth of centimeters, several thermal wavelengths, and characterized by 7 ~ 1000. In the last five ),ears important observations at millimeter and centimeter wavelengths have been made in the Soviet Union by Salomanovich, Troitsky, Kislyakov, and many others. Krotikov and Troitsky (1963) have summarized this work. Within the confines of a linear theory, e.g., one in which the thermal and electromagnetic properties are assumed to be temperature-independent, these data have been interpreted in terms of a homogeneous lunar surface to a depth of several meters, characterized by ~ ~ 350 as opposed to the value of 1000 suggested by the infrared data. Their interpretation of these data apparently is inconsistent with a thin layer of highly insulating material covering a large fraction of the lunar surface, which would exhibit the rapid surface cooling observed in the infrared during an eclipse and during the lunar night. A further important result of this work is an apparent measurement of the thermal gradient in the lunar surface. Krotikov and Troitsky (1963, 1964) have interpreted the observed increase in the mean brightness temperature as measured by several radiotelescopes using the "artificial Moon" call-

608

JEFFREY L. LINSKY

bration procedure (see Krotikov, Porfiryev, and Troitsky, 1961) in terms of a heat flux produced by a level of radioactivity four to six times in excess of that predicted by MacDonald (1959), Levin and Maeva (1961), and Jaeger (1959) under the assumption of chondritic lunar material. However, the measured thermal gradient, as Krotikov and Troitsky have shown, produces a temperature ~ 1000°K at a depth of 60 km and may well lead to a molten lunar interior, if the Moon consists in large part of this mateHal. The observed nonspherical shape of the Moon makes the possibility of a molten interior highly questionable. Thus, recent infrared and radio observations, when interpreted in terms of temperature-independent thermal properties for the lunar surface material, lead to contradictory and disturbing conclusions. To explain the contradiction between infrared and radio observations, one must show that either one or both sets of observations has been interpreted incorrectly; or, there must be a mechanism whereby the lunar material can exhibit more insulating properties (7 large) at the surface during lunar nighttime, while at the same time the material exhibits a significantly smaller thermal parameter at centimeter and meter depths beneath the surface, where the observed radio emission originates. Thus, near the surface during the lunar night, the product of Kpe must be an order of magnitude smaller than a few centimeters or meters beneath the surface. Yet at the same time, the radio data apparently do not allow the surface material to exhibit different thermal properties on lhe average over a lunation from the material directly below it. Thus, an increase with temperature of the thermal conductivity and the specific heat are expected. This tentative conclusion has also been suggested by Krotikov and Troitsky (1963) and Muncy (1958, 1963). Prior to this time, models of the lunar surface including temperature-dependent properties have been computed for special cases by Wesselink (1948), Muncey (1958, 1963), Watson (1964), Tyler (1961), and by Tyler and Copeland (1962).

III. MEASUREMENTS AND LIKELY THERMAL PROPERTIES OF POSTULATED :LUNAR SURFACE MATERIALS

The specific heats of many silicates, oxides, and terrestrial rocks have been measured in the laboratory in the temperature range of 70°-400°K. All of these materials described in Goldsmith, Waterman, and Hirschhorn (1958) exhibit a steep rise in thermal conductivity at low temperatures and then a gradual increase past 200°K to typical values of 0.18-0.24 cal gm-'°K -1 at 350°K. Buettner (1963) compared the specific heats of many such materials and concluded that in the lunar range of temperatures c ~ T 1]~, and thus a temperatureindependent specific heat approximation may be justified. Studies of the photometric and thermal properties of the lunar surface have suggested the basic picture of porous or finely divided "dust" materials in which there are relatively few contact points among the solid particles. In such a class of materials the thermal conductivity will be less-probably orders of magnitude less--than the conductivity of the same material as a solid, and radiative transport of energy in the material may play an important role. In order to obtain insight into the manner in which these two modes of ener~o~ytransport may enter into the effective thermal conductivity, one can consider a number of idealized models of the lunar surface material microstructure. In the first model (Case A ill Fig. 1) I consider a very fragile structure, consisting of cubical cells of dimension s, in which no heat is transported along the thin opaque walls, and the thermal flux is carried only by radiation between them. In this case the thermal flux in direction x is given by F(x,T) = ~ T 4 ( x )

-- ~ R ~ T 4 ( x + s)

and the conductivity is purely radiative ~R(X,T) = 4EtR~T3(x)s.

(1)

In these equations eta is the emissivity of the material, corrected for multiple reflections, corresponding to the peak of the Planck

LUNAR

SURFACE

609

MODELS

T

.S

x

T

i CASE A

CASE B

CASE C

CASE D

X

T Fin. 1. Idealized models of the lunar surface microstructure in two-dimensional projection. Case A represents a structure in which the opaque walls are too thin to support appreciable thermal conduction so that the flux is carried by radiation alone. In Case B and Case C the flux is carried by conduction and by radiation in parallel, except that in the former example the solid material is opaque and in the latter it is partially transparent. Case D is an example of conduction and radiation in series. spectral energy distribution at a t e m p e r a t u r e T, and a is the Stefan-Boltzmann constant. In Case B, I consider an idealized cubical "grainy" structure where the flux is carried b y conduction in the grains and radiation between them, but little or no flux is conducted a t the contact points. In this situation, for a n y flux to be transported a t all, radiation must play a role in tending to equalize the t e m p e r a t u r e s on the upper and lower surfaces of each cell. Therefore, the effective conductivity will be a linear combination of a purely conductive and a purely radiative component

~(z,T) = ~ ( x , T ) + ~o(x).

(2)

T h e purely conductive component of the conductivity will be controlled b y the size

of the contact "points" and thus can be orders of magnitude smaller t h a n if the same material were a solid. If we consider the opposite case of a porous medium in which the solid constituent is partly transparent to infrared radiation, depicted as Case C, then the flux is carried in parallel b y conduction and the radiation field, and the effective flux is again a linear combination of the two components. Now, however, the radiation scale length s is a measure of an infrared photon mean-free p a t h for absorption and not of the geometrical microstructure of the material. I n a more complex medium consisting of transparent and opaque materials, Eq. (2) m a y not be obeyed, but until evidence to the contrary is obtained, one would expect the

()]0

JEFFREY

linear relationship of Eq. (2). Given this relationship, one can characterize the relalive importance of the two components b y their ratio at a specified temperature taken here to be 350°K Ra5o = Ka/K~ = 4eIRa(350)aS/~.

There is, however, an idealized microstructure, albeit unlikely, in which Eq. (2) cannot be obeyed. This example, shown as Case D, is of a material in which the flux is carried b y radiation and b y conduction in series rather than in parallel. Conserving flux in the opaque solid material and the spaces in between, one obtains,

X [1 Rsso[a/(a + ¥s)l(T/350) ~

]

(3)

The gross anisotropy of this model renders it unlikely. Few measurements exist of thermal conductivities over the temperature range 100400°K of finely ground silicates deposited under high vacuum conditions. Bernett, Wood, Jaffe, and Martens (1963) obtained thermal conductivities for several particlesize distributions of crushed olivine basalt in the range 4-15 X 10-6 cal cm -~ see-1 OK-~ between --70 ° and +90°C. Measurements for each sample tended to increase with temperature, but the one sample ( - 35 mesh, p = 1.49 gm cm -3) that exhibited the greatest temperature variation can be characterized in the temperature range - 1 0 ° to + 9 0 ° C b y ~ = 2.2 X 10-6 and R~0~---2.

L. L I N S K Y

Buettner 11963) describes measurements performed on crushed basalt powder of 5u grain size that can be characterized by ~ 9 X 10-6 and R350°~ ~ 2.5. These and subsequent data are summarized in Table I. Watson (1964) has measured the thermal conductivities of dry powdered silicates aI~ 10-5 to 10-6 torr between 150 ° and 350°K. His data are well described by the functional form (2) with the radiative and conductive terms a function of particle size. In particular, the conductive component was found to be independent of composition and inversely proportional to particle size. The radiative component consisted of a constant term corresponding to transmission through small grains and a term linear in particle size corresponding to radiation between large particles. More recently Wechsler and Glaser (1965) have summarized the measurements of the thermal conductivities of porous rocks and powders. The data for rock powders measured at pressures near 10-~ torr fall in the range of 5 X 10-6 < K(T) < 10 X 10-6 cal cm -~ sec-~ °K-L T h e y cite thermal conductivity measurements that increase 30~/c for solid pmnice (between 223 ° and 323°K), 50% for basalt lava (between 220 ° and 265°K), and 30% for basalt powder (between 221 ° and 331°K). Their first preliminary data on basalt powder (104-150 g particle size) at the high vacuums of 5 X 10-~° torr exhibit a very significant increase of 40-50% between 280 ° and 330°K. In this temt)erature range K~~ 6 X 10-7 and Ra~0 -~ 10, but the data are widely scattered and Ras0 m a y 1~ot be well determined.

TABLE I SUMMARY OF I{ELEVANT MEASUREMENT~ OF THE THERMAL PROPERTIES OF POSTULATED LUNAR MATERIALS p (gm

B e r n e l t ctal. (lf,(!3) Buettner (1963) W a l s o n 11964) Watson (1964) W a t s o n (19C4) W e c h s l e r & Gla~'er 11965) F o r -), = 1(1110, c = 0 . 2 For7 = 350, c = 0.2

cln -3 )

1.49 2.61 1.50 I. 50 I . 50 I .1)1) 1.(10

Grain size (microns) 20-2I)0 5 30 100 301) 104-150 0 (I

~(,°K .,. (10 -6 cal cm -t °K-~ sec ~) 6.6 31. 4 . :I 6.7 14. 6.6 51) 41.

2.2 91) 1.4 (I. 67 II. 2 4 (). !i 5.~ 41.

Ra~o 2. 2.5 2 9 57 tO. 0 0

61l

LUNAR SURFACE MODELS

These data as summarized in Table I are far from conclusive, but they do make it imperative that radiative conductivity be considered. Unfortunately, other effects such as possible changes in density, adhesion, or contact resistance with temperature may be important and may affect the thermal properties. For this reason I have calculated models also for a general power law approximation to the conductivity over the lunar range of temperatures K(x,T) = K0(x)T".

(4)

IV. COMPUTATIONAL PROCEDURES

Computations are described of the surface temperature T(O,t) and internal temperature distribution T(x,t) during a lunation and an eclipse in a plane-parallel medium in which the thermal and electromagnetic properties are assumed only to be a function of depth x beneath the surface. The surface material has been assumed opaque in the middle infrared (5-40t t), as suggested by Launer (1952), but partially transparent in the radio region. The heat-conduction equation

pc(x,T)

OT(x,t) = Ot

~(x,T)

Ox

J

(5)

OT(x,t) ] e,,aT4(O,t ) Ox J~o = -

Model Type /--Temperature-independent thermal properties ~(x,T) = K(x) c(x,T) = c(x) ~°(x)

= ~3~Q(x) =

[~(x)p(x)c(x)] - ' 2

Model Type / / - - R a d i a t i v e and thermal conductivity ~(x,T) = c(x,T) = "¢c(x) = ~35Q(X) =

Kc(x) + 4ezaaT3(x,t)s(x) c(x)Tb(x,t) [~(x)p(x)c(x,350°K) ]- ' n [K(x,350°K)p(x)c(x,350°K)] - m

Model Type I I I - - P o w e r law approximation to the thermal properties

subject to boundary conditions

~(x,T)

the surface and a depth of 4 or 5 thermal wavelengths, and for at least 4000 times during a lunation. Care was taken to damp out errors in the assumed initial temperature distributions, so that the internal temperature distributions should be correct to E l ° K , at least to a depth of one thermal wavelength. Computations have been performed for three different representations of the thermal properties in order to determine the effects of nonlinearities in the heat-conduction equation upon the observable thermal emission.

~,x(~,,,t),

(6)

T(x,t) = constant as x -~ large,

(7)

and

is solved by difference equation techniques ou an IBM 7094 computer. In Eqs. (5), (6), and (7) we define ~rR as the mean emissivity at the wavelengths corresponding to the peak of the Planck distribution at lunar temperatures; EB, as the bolometric emissivity for solar illumination; and I($,~,t), as the insulation at lunar rectangular coordinates (~,~), and time t. The complete difference equations and the lnethod of their solution are fully described in Ingrao, Young, and Linsky (1965) and Linsky (1965). All of these computations were made for 20 discrete depths between

K(x,T) = Ko(x)T°(x,t) c(x,T) = Co(x) Tb(x,t) 3,35o(x) = [K(x,350°K)p(x)c(x,350°K)] -11~'. Solutions of the heat-conduction equation in homogeneous media are most easily expressed as functions of the thermal parameter % when the depth variable is expressed in units of the thermal wavelength,

llr = PK/ (~rpc)~12,

(8)

the depth at which the amplitude of the first harmonic in a Fourier expansion of T(x,t) is reduced to e-1 of its surface amplitude. Here P is the lunar synodic period. (See Wesselink, 1948 and Carslaw and Jaeger, 1959.) However, if one wishes to consider a multilayer model, it is often more convenient to consider the physical depth x as the independent variable. In this representa-

612

JEFFREY

tion, solutions of the heat-conduction equations are functions of the diffusivity = ~/pc.

(9)

For each model computed, the specific heat has been assumed to be 0.20 cal gm -~ °K -~ or this value at 350°K when temperaturedependent, and unless specified, the density has been assumed to be I gm cm-3. Thus one needs only to specify K(x,T) for each model and one may characterize each in snore familiar terms by the value of ~3~0(x)= [~(x,350°K)p(x)c(x,350°K)]. Each solution will be invariant to multiplying Kand p by a constant factor with "Y350being reduced by that factor. For these calculations, a value of 0.88 has been assumed for eB and ex~. The former value has been estimated from the albedo data cited in Harris (1961) while the latter is in agreement with measurements of 40 and 400 ~ quartz sand by Burns and Lyon (1963) and measurements of assumed lunar powders in the 7-14 ~ wavelength region by Van Tassel and Simon (1964). Krotikov and Shchuko (1963) have shown that the nighttime surface temperatures, which most conclusively differentiate one model of the surface from another, depend only very weakly upon e8 and e~R for models with temperature-independent properties, and the same is true for the models under consideration. In agreement with Allen (1963), I have assumed a value of the solar constant of 1.99 cal cm-~ rain -~, corresponding to a nonrotating blackbody equilibrium temperature of 395°K at 1 a.u. Deviations from a circular orbit considered by Tyler (1961) are of secondary importance and have not been taken into account. Infrared brightness temperatures characterize the radiation from a grey surface of emissivity eXR as observed through a wideband 8-14 ~ filter. The procedure for deriving these quantities is described ill Ingrao, Young, and Linsky (1965). These brightness temperatures will be subject to small corrections if the lunar spectral emissivity is not constant in the 8-14 ~ region, as Burns and Lyon (1962) suggest, but they will vary significantly only at high temperatures for changes in the effective era. The angular

L. L I N S K Y

dependence of e,a, as measured by Geoffrion, Korner, and Sinton (1960), and a possible transparency of the surface material in the infrared region, as discussed by Buettner (1963), have not been considered. If one assumes homogeneous electromagnetic properties, e.g., no significant change in porosity or chemical composition with depth, and no scattering of thermal radiation at radio wavelengths in the medium or at the surface, the radio brightness temperature TR(h,t) for wavelength k in the Rayleigh-Jeans approximation, shown by Piddington and Minnett (1949), is

TR(k,t) = (1 -- R) fo "~ T(x,t) X exp (--k~x sec 0~,)k~ sec 0i~ dx.

(10)

Here R is the surface reflectivity; kx, the electromagnetic absorption coefficient at the observed wavelength k; and O~n, the angle with respect to the normal made: by an observed ray when in the medium. The method of evaluating Eq. (10) numerically will be described below. In a dielectric, kx will be of the form

kx = ko/~,

(11)

but if lattice vibrations are important, as Sinton (1962) suggests at the short millimeter wavelengths, this relation will not be valid. A value for R may be obtained either from the dielectric constant and the Fresnel laws or from radar measurements. For these computations R is assumed to be 0.05 at the center of the disk corresponding to a dielectric constant of 2.5, in close agreement with the values of 2.8 based on radar measurements of Evans and Pettengill (1963) and Rea, Hetherington, and Mifflin (1964). If the dielectric constant of the surface material is as low as 1.5, as suggested its Krotikov and Troitsky (1963) or in the range 1.7 to 1.8 according to Hagfors eta{. (1965), then R ~---0.01 or 0.02, respectively, and all computed radio brightness temperatures should be increased. For the computation of eclipse and lunation surface temperatures, it is not important to know accurately temperatures at depths ill excess of a few thermal wavelengths. However, in the radio region of the spectrum

613

LUNAR SURFACE MODELS

one m a y be observing radiation emitted from d e p t h s on the order of 10 or more times ),, and thus one must know temperatures at depths in the order of meters. The value of T(x,t) more than a few centimeters beneath the surface will be independent of time and m a y be easily computed, assuming temperature-independent thermal properties, from the average value of the surface temperature over a lunation and an assumed mean thermal flux. In the general case of temperature-dependent properties, the determination of T(x,t) beneath the surface is more complex.

and

(K(x,T)[OT(x,t)/Ox]) -- (F(x,t)) = constant.

(18) In the special case of no ivternal heat sources and therefore no net thermal flux, this constant will be zero. Thus, for a stratified medium in which the thermal conductivity is depth-dependent but not a function of temperature,

(T(x,t)) = ( T ( x -- O,t))

= (T(x = O,t)) + (F(x,t))

OQ(x,t)/ot = OF(x,t)/Ox,

(12)

where F(x,t) is the thermal flux

F(x,t) = K(x,T)[OT(x,t)/Ox],

(13)

and Q(x,t) is the heat content per unit volume of the material

Q(x,t) = fo T(~'t) p(x)c(x,T) dT.

(14)

Taking the time average of both sides of Eq. (12) over a lunation of synodic period P, one obtains

1 f P OQ(x,t) dt = 1 f P aF(x,t) Jo Ot PJo ~ dr.

(15)

The left-hand side of this equation m a y also be written as

0 1 f P OQ(x,t) ot (O(x,t)) = P .Io O'----[--dr,

(16)

where the symbol ( ) means time average over a lunation. Since the solar constant and presumably any internal heat sources in the lunar surface show no secular variation, and the Moon never deviates far from 1 a.u. from the Sun, o(Q(x,t))/ot must be zero. Therefore,

O ~(x,T

) OT(x,t)~ ~x / = O,

(17)

K(X')

(F(x,t)) ~ O.

if

V. VARIATION OF AVERAGE TEMPERATURE WITH D E P T H IN THE LUNAR SURFACE

The heat-conduction equation (5), m a y be written in the form

(F(x,t)) = 0

if

(19)

However, if the thermal conductivity is temperature-dependent, the mean value of the temperature m a y increase or decrease with depth even if (F(x,t)) is zero. This fact has been noted b y Muncey (1958) and Krotikov and Troitsky (1963). For (F(x,t)) = 0, and assuming K(x,T) = Ko(x)To(x,O,

(20)

one obtains

(T'+1(x,t)) = constant -- (Ta+l(x = O,t)) if a ~ - 1 , or (21) (log [T(x,t)]) -- constant =

(log[T(x=0,t)])if

a =

-1.

Now one writes T(x,t) as a Fourier series at each depth

T(x,t) = T o ( x ) +

~

T~(x)

X cos [hoot - ~#n(x)], (22) where Tn(x) and Cn(x) are the amplitudes and phase lags relative to insolation, and o~ = 2 r / P is the lunar angular frequency of apparent rotation. Substituting Eq. (22) into (21), one obtains for integral positive values of a

12

To'(X) = (T2(x,t)) -- 5

Tn2(x)

n~l

if

a = 1,

(23a)

t) i 4

JEFFREY L. LINSKY

To3(x) = (T3(x,t))

3T0(x)2 ~

T,,"(x)

if a = 2

(231))

and for nonintegral positive values the infinite series

To,+l(x) = (T,+l(x,t) } X ~

T,,%r) +

. . .

a(a + 1) T0,_l(x) 4 if" a > 0.

(23c)

the night it must be large and positive. The average gradient is thus positive and the average temperature increases with depth. This effect becomes more pronounced near the surface, where the amplitude of temperature variations is large, as is described by Eq. (23). According to this argument, when the thermal conductivity decreases with temperature, the average temperature must decrease with depth log~ [To(x)] : (log~ [T(x,t)I}

n=l

Since the amplitudes T,,(x) are real and decrease monotonically with depth and since the first term on the right-hand side of each equation is a constant, Eqs. (23a), (23b), and (23c) all exhibit solutions in which the average temperature To(x) increases with depth, asymptotically approaching a limiting value that depends on this constant. This behavior is exhibited in Fig. 2 where the increase in To(x) is given by AT = To (asymptotic) -- To(x = 0). This increase in the average temperature with depth in the absence of internal heat sources can also be explained in terms of the following physical argument. To conserve flux at each depth over a lunation, as much heat must be conducted inward at this depth during the lunar day, as is conducted outward during the lunar night. By lunar night and day I mean those parts of a lunation in which the thermal gradient is negative and positive, respectively, at the depth under consideration. When the conductivity is temperature-independent, but not necessarily depth-independent, flux conservation requires the mean thermal gradients during (lay and night to be equal but of opposite sign. Thus the average temperature must 1)e independent of depth. However, when the conductivity depends upon temperature, the mean gradients dm'ing the day and night cannot be equal. In particular, if the conductivity is proportional to T ~, then flux conservation requires that the average value of T 3 times the temperature gradient during the day be cancelled by the same quantity during the night. Therefore, during the day the telnperature gradient is small and negative, 1)111 during

l ~ T,,~(x) To2(X) if

+-~

et = - I ,

(24a)

and

To-.+%v) = (T-.+~(x,t)} +

Tj(x)

×

Td+l(x)

if

a(l -

a #

a)

-I,

tt =1

a < 0

(24b)

Now To(x) decreases with depth approaching limiting values as shown in Fig. 2. For a two-term expression

K(x,T) = Kc(X) + Ko(X)T~(x,t), similar fornmlas with similar dependencies upon depth may be obtained. In particular, fora= !

To(x) = (T(x,t)} + ~o(x) (T'-(xJ); 1 [ T 0 2 ( / ) + ~T-~j(x) 1. (25) 2 '2 n = [

When (Fix,t)} is not zero, a further increase of To(x) with x is superimposed upon the previous curves. For examI)le, if a is positive and ~(x,T) = ~0(:~')T°(x/),

the analog of Eq. (21) contains an additional term

(V:+%c,t)) = ((~ + 1)(E(x,t)).

fo""~o(xt) dx~

+ ~T°+'(x = 0,t)),

(26)

LUNAR SURFACE MODELS 5O

'

'

'

615

I

'

I

o 40

30

20

~

MODEl

/

PARAMETERS

::: ::::::::::

10

,

i

,

I

~

i

a = l, b =I a=l,b

o =Trr

a =-l,b =0

2

500

0

x = Trr * = 1"I"r

I iO00

J

,

=O

,

I 1500

_1 Y3SO

= (KpC) 2

FIG. 2. The difference AT between mean surface temperature and the asymptotic value of temperature beneath the surface, assuming no net thermal flux, plotted as a function of thermal parameter 785odefined at 350°K. Model Type II includes radiative conductivity, and Model Type I I I includes a general power law approximation to the temperature dependence of thermal conductivity and specific heat. AT is solely a result of nonlinearity introduced into the heat-conduction equation and is zero when thermal properties of the medium are assumed to be temperature-independent. a n d n o w t h e g e n e r a l s o l u t i o n of Eq. (23c) becomes

Toa+l(x) = (a + 1)(F(x,t)} fO~ ~o(X') dJ + (T'+J(x =

0,t)}

a(a + 4

1)

X To"-l(x) ~ Tn2(X) + . . . . ~'t = 1

(27)

T h u s t o t h e p r e v i o u s s o l u t i o n s a d e p t h dep e n d e n c e is a d d e d t o To(x) r o u g h l y p r o p o r t i o n a l t o x ['/(a+l)]. I n t h e a b s e n c e of i n t e r n a l h e a t sources, t h e difference b e t w e e n t h e m e a n v a l u e of the temperature at the surface and its asymptotic value, reached at that depth at w h i c h t h e t e m p e r a t u r e does n o t v a r y o v e r a l u n a t i o n , can be a s large as 50°K. T h i s t e m p e r a t u r e difference A T increases w i t h t h e

616

J E F F R E Y L. LINSKY

value of a and decreasing conductivity, but is practically independent of any temperature dependence of the specific heat. In Fig. 2 curves are plotted of the temperature difference AT, assuming (F(x,t)) is zero. As expected from Eq. (18), any temperature dependence of the specific heat plays an unimportant role, ~whereas the ratio of radiative to conductive flux at 350°K, R3~0, or the conductivity temperature exponent are the important parameters. For two-layer models, AT is intermediate between that characterizing the upper and lower layer materials. When the upper layer depth is about one-third of l~r, AT is the mean of that computed for each layer separately. For the present I will consider the thermal wavelength lit for temperature-dependent models to be the value of PK/0rpc) ~/2 at the mean surface temperature. In the simplest case of temperature-independent thermal properties

f ~ dx' (T(x,t)) = (T(x = O,t)) + (F(x,t)) Jo ~(x---)" (2s) For (F(x,t)) = 2.5 X 10-Tcal cln-2 s e c -1, a s suggested above, (T(x,t)) increases over a distance of 4l~T by 0.9°K for ~/ = 1000 and 0.15°K for -y = 350. Similar small increases in (T(x,t)) characterize temperature-dependent models. Thus the curves of AT vs. "~350io Fig. 2 are essentially unmodified. Therefore, in evaluating the integral in Eq. (10), one can assume (F(x,t)) -- 0 to a depth of 4/lr and use the temperature distributions T(x,t) computed by adjusting T(x = 411T), the temperature at the deepest point considered, such that flux conservation is obeyed. Beyond this depth the temperature is independent of time. Assuming a constant thermal flux at these depths,

F = (F(x,t)), ~ T h e s t a t e m e n t s in K r o t i k o v a n d T r o i t s k y (1963, 1964) t h a t a t e m p e r a t u r e d e p e n d e n c e of t h e specific h e a t c a n i n t r o d u c e a c h a n g e in m e a n t e m p e r a t u r e w i t h d e p t h are n o t s u p p o r t e d b y c o m p u t a t i o n s for a m o d e l in w h i c h a = 0 a n d b = 1; a n d t h e a b o v e theoretical considerations do n o t allow for a n y s u c h variation.

the temperature for Model Type III will be given by

T(x) = {(Ta+l(x = 0,t)) + (a + 1)F fo ~ ~dx' ,

~I/(o+I) x > 4/1T (29)

and for Model Type II by the solution to the equation

T(x) + .

o'~z:T4(x) Kc(x)

= F

fo x dr'

~c(x')

s ( x ) (T4(x = O,t)) + 4l~T. (30) Radio brightness temperatures T~(~,t) have been obtained by inserting computed values of T(x,t) and T(x) at large depths into Eq. (10) for the eight models consistent with infrared observations as described below. These radio temperatures evaluated at the subterrestrial point for 30 times during a lunation have been fitted to a three-term Fourier series, 3

TR(~,t) = TRo(~) + )~ T~,,(~) n=l

× cos [n~t - ~n(x)],

(31)

using a least-squares procedure. The values of Ta0(h), Tin(h), and ~1(),) are listed in Table II. VI. AN INTERPRETATION OF MEAN RADIO BRIGHTNESS TEMPERATURES

Observations of the mean temperature of lunar thermal radiation cited by Krotikov and Troitsky (1963) were obtained with telescopes of low angular resolution compared to the lunar diameter. They have been modified by the correction procedure of Krotikov (1965) to be applicable to the center of the disk, assuming a dielectric constant of 2.5, and are plotted in Fig. 3 along with the values of TR0(k) for the eight computed lunar surface models consistent with infrared lunation measurements. The effect of temperature-dependent thermal properties should not subject this modification procedure to systematic errors that

Kasehenko, Losovsky, a n d Salomonor i c h (1961) 3.2 cm, 6' Mayer, McCullough, and Sloanaker (1961) 3.2 cm, 9 ~

8 ram, 2 p

.

195

.

211 .

-. .

219 196 228

223

--

T~o 0, )

Observations (wavelength a n d beam width)

265.5 274.0 279.6 263.0 272.0 266.4

247.I 266.9

229

217.6 221.5 225.3 217.8 223.2 222.8

209.3 210.8

~'c = 1075 3'c = 1075, 7c = 250(X > 30 em) ((F) = 4 . 8 X 10 -6 calem ~ see -l) 7 ~ = 885, R = 1, b = 0 7a5o = 8 1 0 , R = 2 , b = 0 ~s6o = 750, R ffi 3, b = 0 ~so = 670, R ~ 1, b ffi 1 ~ o = 850, a ffi 1, b = 0 "r~bo = 625, a ffi 1, b ffi 1

Low a n d Davidson (1965) 1.2 ram, 3'.9, 1'.0 Fedoseyev (1963.) 1.3 mm, 10' G a r y et al. (196,5) 3.3 ram, 2'.9 K i s l y a ko v a n d Salomonovich (1963) 4 ram, 1'.6 Salomonovich and Losovosky (1962)

I[ II III III III III

i I

Model p a r a m e t e r s

TRo (50 cm) (°K)

TRO

(1.2 r a m ) (°K)

.

.

.

.

.

--

--

.

.

0.097 0.101 0.097 0.062 0.090 0.062

0.096 0.096

.

.

. .

.

.

.

0 555 . . .

--

0.564 0.556 0.538 0.568 0.523 0.538

0.578 0.572

ko Tin(1.2 m m ) (cm -l) ~TRo(].2 m m )

16

16.1 15.4 15.6 16.2 15.9 15.4

16.1 16.1

27.3 27.3 27.2 26.3 26.2 27.3 26.8 26.0

0.385 0.379 0~371 0.365 0.349 0.374 0.335 0.355

(3.3 ram) (deg)

AND C O M P U T E D M O D E L S

TABLE II

(1.2 m m ) Tin(3.3 m m ) (deg) 'TRo(3.3 ram)

RADIO D A T A

.

.

--

0.377

0.331 0.327 0.312 0.335 0.298 0.318

0.345 0.341

.

TR,(4 m m ) ~TRo(4m m )

. .

--

27

29.4 28.4 28.3 29.7 29.0 28.3

29.5 29.5

. .

.

.

0.189

.

.

0.205 0.203 0.193 0.207 0.184 0.198

0.215 0210

(4 ram) Tin(8 m m ) (deg) 'TRo(8 m i n i

. 30

36.5 35.5 35.3 36.7 35.9 35.2

36.5 36.5

.

(8 m m ) (deg)

0.061

0.076

--

.

0.064 0.060 0.057 0.061 0.054 0.059

0.064 0.064

TRo(3.2 cm)

Tm (3.2 era)

44

45

--

44.3 43.4 43.3 44.5 43.4 43.4

44.4 44.5

¢1 (3.2 cm) (deg)

¢2

J E F F R E Y L.

61~

LINSKY

280

270

260

250

:It .... .,,< --o

240

,._= 250

f-_-_V

oo.n

.....

~ . -

220 I ~

~

/ +

~ " 210

200 0

I

I

I

I

I0

. . . .

!

20

. . . .

I

30

o : n

Y350 = 750,R350 = 3, b =0

• = II

)'350= 810, R350=2, b =0

,=

7"350= B ~ , R:350 = I, b = 0

=

"n

. : n

Y35O=95O,R35o:l,b

. = Trr

7"350=850.0=1, b ~-0

= TIT

X350 = 625, o ,= I, b = I

. . . .

I

40

. . . .

=1

50

k[cm)

Fro. 3. Curves showing computed values of the mean component of radio brightness temperature at the center of the disk as a function of wavelength. Data points are those tabulated in Krotikov and Troitsky (1963), modified to correspond to center of disk by use of parameters computed by Krotikov (1965) and a dielectric constant of 2.5. Computed temperatures include surface reflectivity loss of 5%. Net thermal flux of 3.4 >< 10-7 cal cm -~ sec-~ has been assumed, for all models but one, so as to produce a rate of increase of radio brightness temperature between 3.2 and 50 cm commensurate with the data. A flux of 4.8 X 10-~ cal cm-2 sec-~ was chosen for the two-layer model to satisfy the data between 0.4 and 3.2 cm, and the lower layer thermal parameter was chosen to produce the observed increase in temperature between 3.2 and 50 cm. d e p e n d u p o n X. F o r a dielectric c o n s t a n t of 1.8, a l l of t h e s e d a t a p o i n t s w o u l d be decreased by 1% and the computed radio temperatures increased by 3%. Except where n o t e d , a n i n t e r n a l flux of 3.4 )< 10 -7 cal e m -~ sec -1 h a s b e e n chosen t o fit t h e o b s e r v e d r a t e of t e m p e r a t u r e increase a t w a v e l e n g t h s longer t h a n 3.2 cm. U n f o r t u n a t e l y , a b s o l u t e flux m e a s u r e m e n t s a r e i n d e e d v e r y difficult, a n d it is a c r e d i t t o t h o s e w h o designed t h e " a r t i f i c i a l M o o n " m e t h o d t h a t i t h a s been used. H o w ever, t h i s p r o c e d u r e r e q u i r e s e l a b o r a t e c o r r e c t i o n s for d i f f r a c t e d t e r r e s t r i a l r a d i a t i o n b y t h e o b s e r v e d d i s k (see T s e y t l i n , 1963, 1964) a n d a s s u m p t i o n s concerning t h e effect i v e solid angle of i n t e r c e p t i o n of t h e d i s k ' s d i f f r a c l i o n p a t t e r n b y the E a r t h a n d its

r e l a t i o n to t h e diffraction correction. ~ Since t h e effects of d e v i a t i o n s f r o m t h e s e a s s u m p t i o n s will d e p e n d u p o n t h e d i s k ' s diffraction pattern and therefore the observing wavel e n g t h , a n y s y s t e m a t i c errors in t h i s c a l i b r a tion procedure may be wavelength-depende n t a n d m o d i f y m y conclusions. T h e s i m p l e s t m o d e l s t h a t p r e d i c t a dep e n d e n c e of Tao(X) u p o n )~ c o n s i s t e n t w i t h the absolute mean temperature measurem e n t s in Fig. 2 a r e t w o r a d i a t i v e m o d e l s ('}'350 = 885, R35o = 1, b = 0) a n d ( ~ o = 670, R35o = 1, b = 1); a n d t h e t w o - l a y e r t e m p e r a t u r e - i n d e p e n d e n t m o d e l ( ~ = 1075 u p p e r l a y e r 30 c m d e e p a n d % - 250 lower l a y e r ) . T h e p a r a m e t e r s for t h e l a t e r m o d e l 2 Note Eq. (10) in Tseytlin (1963).

LUNAR

SURFACE

were chosen to exhibit a steep rise in TR0(~,) between 1 and 3.2 cm and a less steep rise at larger wavelengths. If the dielectric constant is smaller than the assumed value of 2.5, one must modify the parameters in these three models to effect good agreement with the data. Choosing a dielectric constant of 1.5 as a plausible lower limit, I have interpolated among the computed models to obtain the estimated parameters shown in Table I I I .

619

MODELS

The above analysis shows that under the assumption of temperature-independent thermal properties, only an increase of the conductivity with depth can account for the form of the mean radio brightness data shown in Fig. 2. No distribution of materials on the surface can produce other than a linear increase with wavelength. However, the mean thermal flux necessary to maintain the observed TR0(X) for the two-layer models is from 15 to 23 times that predicted for

TABLE ]II MODELS IN AGREEMENT WITH M E A N I~AD10 BRIGHTNESS DATA s

Model parameters

I I1 lI

3,~ = lO75 "r3:.0 = 885 78:,o = 670

I lI II

~ = 1075 ~3~0 = 975 ~,60 = 1030

~

,=2.5 = 250(x > 30cm)

R3:,o = 1

b --- 0

= 1 b= 1 ~=1.5 ~ = 460(x >30cm) Ras0 = 1/2 b = 0 R350 = 1/2 b = 1 R3:,o

From these parameters and the data one can place outer limits upon the deduced mean thermal flux passing through the surface material due to internal heat sources. Using the conductivities appropriate to the four radiative models described in Table I I I and a criterion of satisfying six of the eight data-point error brackets between 3.2 and 50 cm, I obtain a value of 2.7 X 10-7 cal cm -2 sec -I for the lower limit, and 4.2 X 10-7 cal cm -2 sec -1 for the upper limit. Corresponding values of the flux for the two-layer nmdels with temperature-independent properties are 3.7 X 10-6 and 5.8 )< 10-e cal cm -~ sec -1, respectively. An alternative explanation of the increase in mean temperature with depth is that this is the thermal gradient between a hot interior and a cold surface. According to the interior models of MacDonald (1959) which exhibit the largest surface thermal gradient, typical values are 0.04°C/meter. These gradients are about three orders of magnitude lower than the radio data suggest and would lead to no measurable increase in the mean radio brightness temperature with wavelength.

(F(x,t))

(mm)

(cal cm2 see -l)

-0.16 0.27

4.8 x 10 " 3.4 X 10-7 3.4 X 10-~

-0.08 0.ll

3.3 X 10-6 3.4 )< 10-7 3.4 X 10-7

chondritic materials. More radioactive materials do exist, but if meteoritic samples m a y be considered a guide, they are less common in the solar system. In addition, there is the problem of supporting the nonequilibrium figure of the Moon, which is aggravated by this high level of radioactive heating. Therefore, the radiative models appear to be the nmre plausible explanation of the data. A further result of this analysis is au estimate of the distance scale of separation of radiating surfaces in the lunar material. This mean effective spacing would correspond to an average pore size in a porous medium or to an interparticular spacing in a grainy or dusty medium. Typical values of 0.1 to 0.3 are suggested in agreement with Wesselink's (1948) original estimate. If the lunar material is partially transparent to infrared radiation, these scale lengths set upper limits to the average pore size or interparticular spacing. This distance scale is an order of magnitude larger than that proposed by Hapke and Van Horn (1963) to account for the photometric properties of the lunar surface.

620

JEFFREY

VII. AN INTERPRETATION OF INFRARED OBSERVATIONS In a n y a t t e m p t to understand the thermal behavior of the lunar surface material, one should consider each method of obtaining information in the light of its inherent limitations and the degree to which the fundamental data on the surface material parameters have been smeared out or convoluted with unwanted information. For the interpretation of radio data on the increase of brightness temperature with wavelength, one m u s t compare absolute measurements t h a t are affected b y side lobes, telescope efficiency, and sky background, and the necessity of independent knowledge of the electromagnetic absorption coefficient /¢~. Radio observations of amplitudes and phase lags of Fourier components of the integrated lunar emission require relative measurements and are thus the more reliable. Infrared data on the surface brightness temperature, b y its very nature, must give the most reliable information of the thermal properties of the lunar surface material at

L. L I N S K Y

small depths short of actual measurement in situ. Assuming t h a t one is indeed observing the surface temperature of an opaque medium, this temperature is affected only by an emissivity factor t h a t is unknown and m a y be temperature-dependent. However, in the 8-14 and 17-22 ~ regions of the spe('trum, one is observing (luring the lunar night in the Wien domain of the Planck spectral energy distribution, where the observed temperatures are very weakly affected b y flux measurement errors a~(l departures from a blackbody. For example, in the 8 - 1 4 ~ window a 12% error in the measured flux or an emissivity of 0.88 produces only a I ° K error in deduced surface temperature for temperatures near 100°K, Different models of the surface m a y exhibit midnight and cold terminator temperatures 10-20°K different, and thus one m a y be able to discriminate among models of the thermal properties a few centimeters beneath the surface b y means of these measuren~ents alone. Eclipse cooling curves yield information of the surface material parameters t~)

r30 o

PARAMETERS

'x

R350=1, b =l R350 = I,b =0 R350 : 2, b = O

IIO R 350 = 5, b =O el=l, b =O

=_... -

a=l,b

=I

k--

70

,

I

200

i

,

.

i

i

600

,

,

,

I

,

I000

i

i

l

1400

i

i

i

1800

I

73SO -" (.'pc) " ~

FzG. 4. The minimum surface brightness temperature at the sunrise terminator on the lunar equator as a function of %~.~,0for the models under consideration.

885,-/, = 1250, 810,-/c = 1400, 750,'y~ = 1500, 670, we = 9 5 0 , 8 5 0 , a = 1, b = 625, a = 1, b =

R R R R 0 1

= = = =

1, 2, 3, l,

b b b b

7~ = 1 0 7 5 ~ = 1 0 7 5 , ~c = 2 5 0 ( x > 3 0 c m ) ( F = 4 . 8 X 10 - s c a l c m -2 sec - ' )

II 735o= I I ~3~o = I I "Y3~o= I I ~,.~o = I I I ~'3r,o = I I I -f3~o =

I I

Model parameters

=0 = 0 = 0 = 1

0.16 0.25 0.32 0.27 ---

---

8 (ram)

24.0 38.1 46.5 26.1 33.1 33.0

0 0

AT (for (F) = O) (°K)

TABLE

IV

219.2 218.7 219.2 219.6 219.4 220.1

219.2 219.4

To(O) (°K)

1.29 1.29 1.30 1.30 1.29 1.31

1.29 1 30

T2(O) T0(0)

3.38 3.32 3.35 3.62 3.43 3.73

3.41 3.41

¢~1 (deg)

INfrAReD COMPUTED DATA

4.17 4.33 4.59 6.66 4.50 7.22

4.14 4.15

lie~, (era)

4.33 4.10 4.22 6.90 4.13 6.67

4.15 4.20

Ila (cm)

4.25 4.22 4.41 6.78 4.32 6.94

4.15 4.17

Lxr (°K)

90.0 89.5 90.1 89.8 89.7 90.3

89.7 90.2

Tm~.,B (°K)

98.3 97.2 98.3 98.7 98.5 99.6

98.1 98.5

T~,a (°K)

3 . 2 0 ) < 1 0 -6 2 . 5 5 ) < 10 -6 2 . 2 2 X 10 -6 5 . 5 4 )< 10 -e 1 . 9 9 X 10 - s 3 . 6 3 X 10 -8

4 . 3 3 X 1 0 -6 4 . 3 3 )< 1 0 -6 3 . 4 6 )< 10 -6

~ l p or ~ / p (cal ore= °K -I sec -1 gin-l)

t~

(~2

JEFFREY

LINSKY

L.

160

\

I

x ~

,,o

I

--.i ......

i',l

= 'fl"

o

- ....

P

\ \

R350 = I, b =0

I

~.

"

~,\\

~-

~

.

~

~ ~_~

"

...."~

~

-" ~

.

~

~ - . ~

7c : I 0 0 0

~ ' - ~

~,.~_~_

~

~ ~_

7c : 1 0 0 0

• ~ 80

7¢:1500

7c : 1 5 0 0

-

1=SUNSET I ;270 =

,

SUNRISE" I 300 °

,

I

,

330"

I 360 =

,

,I

30"

,

I 60*

, ~3,"

PHASE

FIG. ,~. Computed surface temperatures during the lunfl,r night ~re compared for radiative (II) and n . n r~tdiative (I) models. The parameter Ra~o, the ratio of radiative to conductive flux 3~t350°K, is a measure (,[ the importance of radiative energy transport in these models. Phase 360° corresponds to the antisubsolar point. smaller depths, but lunar daytime infrared measurements offer no information on these properties, since the insolation ahnost completely controls the surface temperature. Recently Watson (1964) has been unable to interpret adequately the infrared scans across the evening terminator and into the lunar night obtained b y M u r r a y and Wildey (1964). His models included radiative conductivity and assumed a homogeneous stratified medium. I also have had difficulties in interpreting these data due in part to the complex nature of the observed thermal

emission near the evening terminator. When observations are obtained with a finite angular resolution over a region where the surface temperature and, therefore, the emitted flux is a rapidly varying function of position, and where shadowing and smallscale inhomogeneities at the very surface play an important role, they do not readily lend themselves to an unambiguous interpretation b y means of simple models. On the other hand, when one observes the thermal emission before local sunrise, none of these complications arises.

LUNAR

SURFACE

MODELS

6~

160

MODEL

IPARAMETERS

-. . . .- .

== T t IT

- - - - -

,

o=l,

a=

rr[

~40

b=O

,

b =

120 e

I--

'7350=500

~00

j 735o: > Ioo0 Z35o: 1500

73~o: iO00

.°

TISUNSlET 270*

"!;"350= 1500 I 300 °

J

I

,

350"

,

I 360"

I 30 °

,

I su", 60 °

90"

PHASE

FIG. 6. Computed surface temperatures during the lunar night are compared for temperature-independent models (I) and models (III) in which the thermal conductivity and specific heat are proportional to the powers a and b, respectively, of the temperature. For these reasons I will use as m y primary source of information infrared data on the minimum temperature reached during the lunar night, and choose those models of the surface that are in basic agreement with these data, for comparison with infrared and radio measurements. The only infrared data available with a signal-to-noise ratio sufficient to measure the coldest lunar temperatures are those of Low (1965). He obtained a mean temperature of 90°K for the cold limb, but observed cold areas of < 70°K and hot areas including one > 150°K. Radio emission also varies

from place to place, as Gary (1965) has recently confirmed, but the variation is much less. Taking 90°K to be representative for the surface brightness temperature minim u m T,,.~ and in particular the value for the subterrestrial point, I have calculated models with a range of thermal parameters that are consistent with this measurement. In Fig. 4 calculated cold limb brightness temperatures are presented as a function of ~350 for the model types considered. These brightness temperatures, as mentioned above, are computed for EIR -----0.88 and the 8-14 ~ atmospheric window, but will vary

624

JEFFREY

I I.I --DATA

'

I

L.

'

LINSKY

1

'

I

'

1

OF PETTIT 0940)

t o =74

minutes

MODEL __= 1.0 : --

o.

--=

PARAMETERS

fl

)'350= 1075

[-I

7350= 1 0 7 5 / 2 5 0 , d = 3 0 cm

n

7 3 5 0 = 6 7 0 R350=1, b : I

]]

) ' 3 5 0 = 7 5 0 , R350=3, b :O

....

o //,...-~J~

...o..,o....o...° .o

y/

0 .0

\\

-j

.....

~.

0.8

=]

-- ....

~o

__

Tr/

_

/

o=,., =o iy

o

/

= 6 2 5 , a=l, b =1

/

_L'35° . . . . . . . . .

/

o.,

/

0.6

0.5

o.,

.5

I

,

I

1.0

i

I

2

,

I

3

,

!

4

t/t o

i,'IC;. 7. Theoretical eclipse cooli1~gcurves are compared with tile data of Pettit (1940) for tile Oct(~l)er27, 1939 eclipse and lunar rectangular coordinates (~ = 0., n = +0.17). For this eclipse the duratio~ of penumbral phase to = 74 min, and the duration of umbral eclipse is 139 rain or 1.88 to. only by a few tenths of I ° K for observations in the 17.5-22 ~ window. The models described in Table IV, representing a wide range of parameters, are consistent with this 90°K criterion. The dependence of brightness temperature upon phase during a lunation differs at most by 4°K at each terminator among the models, and, therefore, does not allow for a distinction among t h e m from infrared measurements alone. In Figs. 5 and 6 the variation of surface ternperature T(O,t) with phase is presented for a representative sample of temperature-independent models to show the quantitative effect of the nonlinearities, During the course of an eclipse, the most relevant information is obtained in the umbral phase when the c o n d u c t i v i t y and specific h e a t of the lunar material control

the cooling rate. One can nfinimize, but not elinfinate, the effects of an unknowu surface emissivity by considering the ratio of observed temperatures to the measured pre-eclipse temperature (see Ingrao, Young, and Linsky, 1965). This approach also minimizes errors introduced hy comparing observed temperatures at positions other than the subsolar point with computations made at the subsolar point. Beyond 30 ° il~ latitude or longitude from the subsolar point, these errors computed in Ingrao, Young, and Linsky (1965) become important. In Fig. 7 a comparison is lnade between the measurements of Pettit (1940) and cooling curves for the eight models computed for the circumstances of the 1939 eclipse arid for the area of the lunar surface observed. None of the models fits exactly t)ut, on the

LUNAR SURFACE MODELS

other hand, a better representation could be achieved only by a two-layer model with a very thin upper layer and a lower layer with ~'350somewhat less than 1000. Another possibility would be a composite model with several materials on the surface. Either of these situations m a y indeed be the case. V I I . AN INTERPRETATION OF RADIO OBSERVATIONS

The relationship between observed brightness temperature at radio wavelengths integrated over the lunar disk ~PR(h) and the temperature at each depth, longitude, latitude, and time T(x,O,~b,t) has been discussed in the literature (see Krotikov and Troitsky, 1963 for full references). To review, one can write T(x,O,¢,t) assuming temperatureand depth-independent thermal properties in the form of a Fourier series

T(x,O,C6t) = T0(C) + f f j (--1)~"T,(~b) Xexp[--x\2-aa]

(

X cos n 4 ) - - n 0 - -

J ]'

=

[1 -

X

R(0,¢)]

T(x,O,C6t)kx see r X exp (--xkx sec r dx),

(33)

= I1 -- R(O,¢/)] l T0(~h)

+

n=l

( _ 1)a-T,~(~) (1 -{- 28,-c-osr +-~-~ cos = r) 112

X cos:[n~p

--

nO

--

O,

--

and the direction at the point of reception; and ~(O,~b) = tan-'[6~ cos r/(1 + ~. cos r)], phase shift relative to insolation for the nth Fourier component of the radio brightness temperature. Very often the antenna pattern integrates over a large part of the lunar disk or the whole disk, in which case the integrated thermal emission will be given by TR(h)

(-- 1)--T.(O)~. (1 + 2a,~ + 2($,,2)1/2' (35) n =I

(32)

and therefore the thermal radio brightness temperature at (0,¢~) as r,(x,0,¢,t)

= 2~t/P; r, angle between the normal to the surface

+ (1 -- R±)

(nP'~V2'~

~b,,--x~2a]

nth Fourier component of the surface temperature; P, lunar synodic period; ~ = nP/(2a) 1/2, ratio of the thermal attenuation coefficient for the nth Fourier component to the electromagnetic attenuation coefficient; ~, diffusivity; a~ = (n -- 1)(n -- 2)/2; R(0,~b), surface reflection coefficient;

TR(X) = (1 -- R±)~0To(0)

n =1 ( n P ~ 1/2 ]

[

625

~,(0,~b)]}.

(34)

The notation is that of Krotikov and Troitsky (1963) with slight modifications and has the meaning

To(¢/),i T,(¢/), Fourier components of the surface teinperature; ~., :phase shift relative to insolation for the

where the coefficients B0 and 5~, which depend upon the dielectric constant ~, the antenna pattern, and the observing wavelength X, have been calculated for the case of a single-lobe antenna beam by Krotikov (1965). In their summary paper Krotikov and Troitsky (1963) give strong arguments for a homogeneous surface layer at least to a depth of 31~T, which is 15-20 cm for the models this analysis suggests, based upon measurements of the phase lags and amplitudes of the Fourier coefficients obtained at a number of wavelengths. These data have been interpreted by Troitsky (1962b) to show ~ = 2X. (36) In addition, three independent arguments are given for a thermal parameter % ~ 350 characterizing the surface material of temperature-independent properties. The first argument given is that the infrared midnight temperature T~'d.B(0) measured by Sinton (1960) is 1 2 2 ° 4 - 3 ° K which is consistent

626

JEFFREY

with 350 < %, < 430. This measurement was obtained by use of a pyrometer of unspecified characteristics and differs by a factor of 50 in flux from a 100°K antisubsolar point temperature corresponding to the cold limb data of Low (1965). Sinton's temperature is also in disagreement with the data of Saari (1964) and Low (1964). Secondly, a value of 250 < ~,~ < 450 is consistent with the absolute measurements of the average brightness temperature at 3.2 cm of 2 1 1 ° 4 - 2 ° K . In contradiction to infrared measurements, these radio determinations of the temperature exhibit the same errors as flux measurements, and the "artificial Moon" calibration is subject to errors as cited above. A decrease of this measured temperature by only 5°K would lead to ~,~ = 1000. The final argument is based on an empirical relation between the ratio of the inean to first harmonic amplitude of the radio brightness temperature and the observed wavelength. This ratio M = TROIS'R1, strictly speaking, is of quantities integrated over the disk, but is nearly equal to the ratio at the center of the disk,

L. L I N S K Y

as given by Krotikov and Troitsky (1963) is based upon a dubious infrared measurement, an absolute radio brightness temperature for which small errors greatly affect the conclusions, and an extrapolation procedure t h a t gives ambiguous results. TABLE V ItOMOGENEOUS TEMPERATURE-INDEPENDENT MODELS

T0(0)/TK0)

~c

1.5 1.4 1.3 1.2

400 6O0 900 1500

However, certain features of their analysis m a y be valid despite their conclusions, and even if the assumption of temperature-independent thennal properties is invalid, those relations will be valid for which appropriaW mean quantities can be suitably defined when the heat-conduction equation is considered in its nonlinear form, Eq. (5). One can rewrite Eq. (32) at the subsolar point (¢ = 0, ~b = 0) for the more general case in lhe form

M ~ TRo(O)/Tm(O). The observed data have been extrapolated to X = 0 leading to a ratio of the surface temperature Fourier components To(O)/ TK0) = 1.5, and the relation ~, = 2X. However, the data m a y also be fitted equally well by the parameters T0(0)/V,(0)

= 1.3

~il

2.4(

or

To(O)/T,(O)~= 2.012} Any one of a number of such parameters will also agree with the data presented. But the value of ~,~ for a homogeneous temperature-independent model varies rapidly with To(O)/T,(O), as shown in Table V. Therefore, this method of interpreting radio observations gives very little unambiguous information concerning the thermal properties of lunar surface materials. In short, the value of ~/~ = 350% 4- 20%

It = 1

X cos

n~,t - ¢.

.-1/,,(T

,

(37)

where Un(T) and V , ( T ) are analogous to a thermal wavelength but will in general not be equal. Since U,(T) and V , ( T ) are, in general, related to the temperature in a nonlinear manner and the mean value of the temperature To(x) depends upon depth, these attenuation and phase-lag distance scales also depend upon depth. At each depth considered in the lunar surface and for each computed model, a one-term Fourier series was fitted to the computed temperatures T(x,t) = To(x) -}- Tl(x) cos [~t -- ¢1(x)].

(3S) and the parameters Tl(X) and ¢1(x) have been plotted in Figs. 8 and 9. Although this procedure is not, strictly speaking, a method

LUNAR

'

I

'

I

'

I

'

I

'

SURFACE

I

'

I

627

MODELS

'

1

'

I

'

I

'/d

'

I

'

I

2 2.o

O~

~

o : I"1"7 y350=750, a =I, b = 0

," ° :

, , , , -0

2

4

,

I

6

,

I

8

;

I

;

I0

I

12

14

16

I,B

: 20

I

22

I

24

x(cm)

FIG. 8. Values of the first harmonic amplitudes of the computed temperature distribution obtained by a least-squares procedure at a number of depths beneath the surface are compared with the surface amplitudes for a number of models. In each case an exponential decrease with depth is an excellent approximation, so that one can define a thermal attenuation wavelength ll~ unambiguously as that depth at which the first harmonic amplitude decreases to e-~ of its surface value. of solving for UI(T) and VI(T) independently, the fact that - l o g e [TI(x)/TI(O)] and ¢,,(x) obey a linear relation with depth to a very good approximation implies that U1 and V1 m a y be considered depth-independent at these depths. Beyond these depths temperature fluctuations become negligible. One can define an attenuation wavelength as that depth at which -- loge [T:(lla)/Tl(O)]

= 1,

and a phase-lag wavelength as that depth at which (I)1(/1,)

--

~1(0)

=

1 radian.

These quantities are listed in Table IV. In each case they differ by less than 5%, so that one can consider their average to be the effective thermal wavelength L~r. A similar procedure could be applied to their higher

harmonics. Equation (37) can now be written as

1). ~1

F r o m this, one can derive the same e_quations for Se(X, ck,~b,t), Ta(X,O,O,t), and Tn(;~) as before except that now

~,(h) = 1/L,vkx.

(40)

In Table IV there is also given the ratio of mean surface temperature to amplitude of the first harmonic for those models corresponding to a cold limb brightness temperature of 90°K. Each of the eight models under consideration exhibits a ratio very

628

JEFFREY

L.

LINSKY

/

150 °

I00"

MODEL : ]I 50*

0 I. 0

z

4

s

e

,o

Iz

t4

PARAMETFRS )'c

;750'R350 =l'b=O ='/,50,R550=2,b:O

x = ~

y¢

o = TIT

7"350=750.0 =l,b = 0

e . . "rrt

) , 3 5 0 = 7 5 0 , o = l. b = i

is

le

zo

zz

24

x (cm)

Fz6.9. Values of the first halmonic phase lags relative to surface insolation of the computed temperature distribution obtained by a least-squares procedure at a number of depths beneath the surface are compared for a number of models. In each case a linear increase with depth is an excellent approximation, so that one can define a thermal phase lag wavelength l,~ unambiguous as that depth at which the first amplitude phase lag relative to insolation increases by one radian relative to the surface phase lag. close to 1.30. From this ratio and the set of possible parameters that are in agreement with experimental amplitude ratio M = TR0/Tn, versus ),, we nmst conclude that 61(X) = 2.4h

(41)

instead of 2~. Computations have been made of the radio brightness temperature at the subterrestrial point during a lunation and an eclipse assuming kx = 1/2.4XLn%

(42)

and the computed values of LaT, listed in Table IV for a number of wavelengths and the eight surface models under consideration. These computations apply, strictly speaking, only to the center of the lunar disk and should be compared with data obtained with a resolution of better than 6' to minimize smearing of relevant detail.

Krotikov (1965) presents correction factors to be applied to the measured Fourier amplitudes and phase lags for an antenna pattern with a central lobe of Gaussian shape and arbitrary width, but no side lobes. These factors were computed for an assumed homogeneous temperature-independent surface model. Both of these assumptions m a y not be valid. Levin (1964) has noted that observations with low angular resolution m a y give a false picture of the true nature of the lunar surface material in any one region. In addition, significant variations in the radio emission have been noted over the surface at 3.3, 4, and 8.6 ram. For t h e ~ reasons, I have chosen for comparison the high angular resolution data of Low and Davidson (1965) at 1-1.4 mm, Gary, Stacey, and Drake (1965) at 3.3 mm, Kislyakov and Salomonovich (1963) at 4 ram, Salomonovich a n d Losovsky (1962) at 8 ram, Koshchenko,

'

I

1.6

"'

I

'

!

'

DATA OF LOW AND DAVIDSON ( ]965 = 3J9 BEAM WIDTH r~O BEAM WIDTH

I.B - -

I

'

I

""

I

'

I

'

[

'

'1

'

f 2 rnm MODEL - - = ( ;

PARAMETERS T¢ = 1075 7c = 1075'XC 250 (x>3Ocm)

=

o

r=,

,"

/

.

.y

J IO

08

o.4 lEO"

'

I 200"

,

l .i 240 °

l . 220=

,

.l. 320*

,

l 360"

,

l 40"

i

l eO"

I 120"

,

,

J 160"

, 200=

PHASE r. Fro, 10. Computed r~dio brightness temperatures a t 1.2 m m for the center of the disk are compared with the data of Low and Davidson (1965). The discrepancy betweea the theoretical curves a n d the data at the sunrise terminator (~ ~ 90 °) may be due to shadowing by the rough surface.

'

I

i

I

'

I

I

'

I

'

I

'

I

'

I

'

I

'

3.3 mat

#.B--DATA OF GARY,$TACEY, AND DRAKE {196B) SEAM WROTH2'.9

MODEL

PARAMETERS Yc = 107.5

Yc = 1075'Yc=250(x>30crn)

1.6

)"350= 670, R350=l,b : I I . . . . .

=

14 ....

~\

:----

1-[

~'3SO-885 R,SO- i, b =0

r 11r

~'350=S I0, R3.~O=2, b =0

~ ]~

T350" 750'R350= 3' b =0

=

)3~o85oo=1

S

/~

b =o

i,_¢

1.0

o.6

OA 160"

,

I 200"

,

I 240"

I

I 200 ~

,

I 320*

,

I

I

360*

40"

c

I 80"

,

I 120"

I

I 160°

200"

kHASE

Fz(i. 11. Computed radio brightness temperatures at 3.3 m m for the center of the disk are compared with temperatures interpolated from the isotherms of Gary, Stacey, and Drake (1965). 629

t):~0

JEFFREY L. LINSKY '

I

'

1

'

I

'

I--'

I

MODEL

I

- - :

t I

I

'

1

'

1

'

t

4 mm

--PAT,& OF K:SL'IAKOV AND SALOMC~NOVICH{IqGS) BE,&M ~ D T H i' 6

.....

=

I 1

PARAMETERS ) " = 1075 7c = 1075,7c = 2 5 0 t ' { > 3 O c m ) 7 ~5:.= 6 7 0 , R 5 5 0 = I , b = I

I]

T

)'350 = 8SS, R 350= I ,b = 0 ¥ ~ b 3 : 8 1 0 , R 5 5 0 = 2, b = 0 7~50 = 7 5 0 , R 3 5 0 = 3 , b = 0 7150: B 5 0 , a : l , b : 0

. . . . . .

=

Ill

7 3 5 o : 6 2 5 , {1 : I. b : I

~.S/

o J

C8

"

--

I ~::

~

T -'r i

C6,

,

0 a!60o

I 200 °

,

I

_~_

240 °

I 280 ~

,

~ 520 °

[

I

/

I ,

360 °

i/

1 40 °

,

I 80 °

,

1 120 °

,

I 160"

, 200"

PHASE

FIG. 12. Computed radio brightness temperature a t 4 m m for the center of the disk are compared with the d a t a of Kislyakov and Salomonovieh (1963).

Losovsky, and Salomonovich (1961) at 3.2 cm, and Mayer, McCullough, and Sloanaker (1961) also at 3.2 cm. In each case the data, which apply to the center of the disk, have been renormalized so that the observed and computed mean temperatures agree. By this procedure I am comparing observed and computed relative temperatures that are unaffected by systematic errors and errors in the assumed surface reflectivity. The 1.2, 3.3, and 4 mm data may not be readily characterized by a one-term Fourier series and are presented along with the eight models in Figs. 10, 11 and 12. At each wavelength there is good agreemen, between computations for each model and the data. The main exception to this is the 1.2 mm data at the sunrise terminator. This discrepancy can be attributed to the fact that shadowing effects are important at solar elevation angles below 20 ° . Mean slopes on the order of 10-12 ° on the scale of I meter have been predicted by Rea, Hetherington, and Mifflin (1964) based on radar back-scattering measurements.

In Table II the computed relative amplitudes T~I(X) and phase lags ~bl(X) for th(, first term of a Fourier series are compared with the data at 4 and 8 ram and 3.2 cm. The good agreement at millimeter wavelengths and at 3.2 cm suggests that each of the eight models may be representative of a large part of the lunar surface, but it in no way discriminates among them. Lunar thermal emission has been observed during a lunar eclipse most recently at a variety of millimeter wavelengths by Kamenskaya et al. (1965), at 1-1.4 mm by Low and Davidson (1965), and at 0.8 and 1.5 mm by Baldock et al. (1965). Unfortunately, only the data of Low and Davidson and the 1.2 mm data of Kamenskaya et al. were obtained with suitable angular resolution, 3'.9 and ~ 1 ' , respectively, and under suitable observing conditions so that they could be reasonably interpreted in terms of the computed radiation emitted by a small area on the lunar surface. These data as shown in Fig. 13 disagree

LUNAR

SURFACE

MODELS

631

1

'

' PEN

-

'

LJ.,,J..B./

---

uMBRAL {12130163)

'

YLEJ.,dJ'

'

'

'

'

PEN(12/3D/6

1,0

o

.92

o

o

o

=

o

o

.1:.

,84

x PARAMETERS

MODEL ~_~

.7B

Jl

:

H

7350 = 7 5 0 , R~50= 5, b ; 0

:

II

7350=670,R]50=1,

:

m

r35o=625,

:

m

T 3 5 0 : 8 5 0 , ~: : I, t ~0

t =J

~ =1, L'

I

.68

?'350 =I075

r550=BOS,B55o

, FOR / 1 2 / 1 9 / 6 4 ECLIPSE

=1, i~ =0

; ¢ 3 5 D = B I O , R 3 5 D =2, b : 0 T350=1075/250,

o DATA OF LOW AND DATIDSDN ( ~965 ) FOR ECLIPSE 1 2 / 1 9 / 6 4 3'.9 BEAM WIDTH

d= 3 0 cm

DATA OF KAMENSKAYA

et ol (1965) FOR ECLIPSE ;2130163, 7' BEAM WIDTH

SAME AS ABOVECURVE 12/30/63 ECLIPSE

, D

60

120

180

,]J 24D

,

I ~00

,

, 3BD

t (minutes)

Fro. 13. (3omputed radio brightness temperatures for 1.2 m m at the center of the disk for the total lunar eclipses of December 30, 1963 and December 19, 1964 are compared with the data of Low a n d Davidson (1965) a n d Kamenskaya et at. (1965). The circumstances of these eclipses are sufficiently similar t h a t the two sets of data should agree.

significantly with each other and with predictions of all the models. One may therefore conclude nothing from them at this time. IX. CONCLUSION~ Infrared and radio measurements can at t~st give information about the gross nature of the thermal properties of the lunar surface to a depth of a few tens of centimeters. Each type of data sheds ~ght more or less unambiguously upon some aspect of the problem, but any proposed model of the lunar surface must be in agreement with all of the data. The logical starting point of this analysis was the typical minimum temperature of 90°K for the cold terminator. Eight models of the lunar surface were proposed, including a wide range of temperature-dependent and -independent properties, characterized by 625 < ~'~60 < 1075. A computer program was written to solve the heat-conduction equation for an eclipse and a lunation as well as to compute radio and infrared bright-

ness temperatures for any region of the lunar disk. This program may, with no modification, be applied to similar bodies such as Mars and Mercury or may find application in considering heat transfer for time-dependent insolation problems such as the decay of a comet. Each of these eight models is in agreement with infrared and radio data at high angular resolution on the lunar disk. However, each model predicts a significantly different mean radio brightness temperature dependence on wavelength. To the extent that the intrinsically difficult precision measurements obtained in the USSR are free from systematic errors that depend on wavelength, two models including significant radiative energy transfer during the lunar daytime appear to be the most plausible. These two models suggest radiative scale lengths of 0.16 and 0.27 mm, respectively, for a dielectric constant of 2.5, or 0.08 and 0.11 ram, respectively, for a dielectric constant of 1.5. Under the assumption of an

(~32

.J~;~~~EY L. LINSKY

opaque material in the infrared, these scale lengths are representative of particle sizes. On the other hand, if the material is partly transparent in the infrared, which is likely for particle sizes on the order of or smaller than a wavelength, these scale lengths may be interpreted as infrared photon mean free paths. In either case one would expect, a porous or grainy medium. This deduction is supported by the radar depolarization measuremenIs of Hagfors et al. (1965), which suggest a tenuous surface layer at, least 20 cm in depth. Hopfield's (1966) interpretation of visual negative polarization a~ small lunar phase angles suggests particle sizes of 5-10/a. This range of sizes may not be in conflict with that, obtained above, since small particles are etfieient polarizers whereas large particles and separations enhance radiative transfer. Lunar surface materials could thus consist of a wide range ot' particle sizes.

A porous surface material does not necessarily imply a covering of "(lust" or a surface shattered by meteoric impact and covere(l b y debris. If large regions of the lunar surface consist of uncovered lava flows, as Kuiper (1965) suggests, it is plausible to assume th(, physical nature of the lunar surface material is that produced by an originally molten material that upwelled and solidifie(I in a vacuum. Recent laboralory experiments by Dobar, Tiffany, and Gnaedinger (1965) have verified that a porous medium is produce(l when molten silica is allowed to upwell in a vacuum. This material is of low density, reproduces lunar photometric era'yes, and discolors when irradiated under condilions simulating the solar wind. The agreement between in(lependent infrared, radio, radar, photometric measurements, and laboratory data suggests that the models presented here may be representative on a gross scale of much of the lunar surface. ACKNOWLE DGMENT,~ This work was sponsored by the National Aeronautics and Space Administration raider G r a n t No. NsG 64-60. The author wishes to thank Professor Fred Whipple, Professor Charles Whitney, Professor Carl Sagan, Dr. James Pollack, and Mr. Richard Munro for very helpful discussions of certain theoretical

aspects of this paper. Tile author also wishes (o t h a n k Professor William Liller, Chairman of the Harvard College Astronomy Department, for the allocation of some of tile c()mputer time required by this report. Acknowledgment is also due to Dr. Owen Gingerieh for his assistance in debugging the coltlpu( er program. The autlmr is indebtcd t,) Professor l)onald }1. Menzel, Principal Investigator, and Hector C. Ingrao, Co-Investigator of this project, for their support and eneouragen/eul (ff this research. [{EFERENCES

ALLEN, C. ~;. (1963). "Astropl~ysical Quantities." Athlone Press, London. BALDOCK,R. V., I~OSTIN,J. A,, CLEGG,P. E., EMERY, R., GArrSKEIm, J. N., AND GFaR, A. E. (1965). Lunar eclipse observations at 1 m m wavelength. Astrop]~y,s', J. 141, 1289. B~:az~E,r'r, E. C., WooD, H. L., JAFI~e:, L. D., .~N, M.aRTENS, }I. E. (1963). Properties of a simulated lunar material in air and in vacuum. Am. hi.st. Aero~io~l. Astro~latd. J. 1, 1402. BVETa'NE~, K. J. K. (1963). The Moon's first del'imeter. Planclary Sp(lce ~ci. 11, 135. B~.~R~'S, E. A., AND LYON, R. J. P. (1963). Paper presented at the Lunar Surface Materials Conference in Boston. BV~tNS, E. A., AND LYOn', R. J. P. (1962). Errors in the measurelnent of the temperature of the Moon. N~tt~o'c 196, 463. CansLaW, H. S., .~NO J:lEaE~, J. C. (1959). "Conduction of Heat in Solids." The University Press, Oxford, England. I)OBAR, W. I., TIFFANY, O. L., AND GNAED1N(IER, ,l. P. (1964). Simulated extrusive magma solidifieation in V'Icuunl. [CflFtt.q 3, 323. EVANS, J. V., AND PETTENG1LL, G. H. (1963). Tile scattering behavior of the moon at wavelengths of 3.6, 68, and 784 centimeters. J. Geopbys. R<'.~.

68, 423. FFOOSEYEV, L. N. (1963). Lunar and solar emission at 1.3 m m wavelength. Radiofizilca 6, 655. GAIcY, B. (1965). Results of a radiometric Moon mapping investigation at 3 millimeters wav(,length. Jet Propulsion Laboratory, in press. GAaY, B., STatics, J., aND Da,~K~, F. D. (1965). Radiometric mapping of the M o o n at 3 millimeters wavelength. A.strophys. J. Suppl. Nor. 12, 239. GEOFFmO~', A. it., KOR>;ER, M., ANn SINTON, W. ~1. (1960). Isothermal contours of the Moon. Lowell

Obs. Bull. 5, 1.

(~OLDSMITI-I,A.. WATERMAN, T., aND HIRSCttHORN, H. (1958). "Thermophysical Properties of Solid MateriMs." W A D C Te(!h. Rept. 58-476, Armour Research Foundation.

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633

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